Geodesic
On a method for approximate solution of a mixed boundary value problem for an elliptic equation
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 23 (2021) no. 1, pp. 58-71.

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A mixed boundary value problem for an elliptic equation of divergent type with variable coefficients is considered. It is assumed that the integration region is a rectangle, and the boundary of the integration region is the union of two disjoint pieces. The Dirichlet boundary condition is set on the first piece, and the Neumann boundary condition is set on the other one. The given problem is a problem with a discontinuous boundary condition. Such problems with mixed conditions at the boundary are most often encountered in practice in process modeling, and the methods for solving them are of considerable interest. This work is related to the paper [1] and complements it. It is focused on the approbation of the results established in [1] on the convergence of approximations of the original mixed boundary value problem with the main boundary condition of the third boundary value problem already with the natural boundary condition. On the basis of the results obtained in this paper and in [1], computational experiments on the approximate solution of model mixed boundary value problems are carried out.
Mots-clés : elliptic equation, Sobolev spaces
Keywords: mixed boundary value problem, embedding theorem, approximation, convergence of approximations, difference scheme, iterative method, grid method. 
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M. È. Fairuzov; F. V. Lubyshev. On a method for approximate solution of a mixed boundary value problem for an elliptic equation. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 23 (2021) no. 1, pp. 58-71. http://geodesic.mathdoc.fr/item/SVMO_2021_23_1_a3/

[1] F. V. Lubyshev, M. E. Fairuzov, “Approximation of a mixed boundary value problem”, Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 20:4 (2018), 429–438 (In Russ.)

[2] K. Rektoris, Variatsionnye metody v matematicheskoy fizike i tekhnike, Mir Publ., Moscow, 1985, 590 pp. (In Russ.)

[3] V. P. Mikhaylov, Differentsial'nye uravneniya v chastnykh proizvodnykh, Nauka Publ., Moscow, 1976, 391 pp. (In Russ.)

[4] O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973, 408 pp. (In Russ.)

[5] David Gilbarg, Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order Second Edition, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1983 | MR

[6] G. I. Marchuk, V. I. Agoshkov, Vvedenie v proektsionno-setochnye metody, Nauka, M., 1987, 416 pp. (In Russ.) | MR

[7] A. A. Samarskij, R. D. Lazarov, V. L. Makarov, Raznostnye skhemy dlya differencial'nyh uravnenij s obobshchennymi resheniyami, Vysshaya shkola Publ., Moscow, 1987, 296 pp. (In Russ.)

[8] A. A. Samarskii, V. B. Andreev, Raznostnye metody dlya ellipticheskikh uravnenii, Nauka, M., 1976, 352 pp. (In Russ.) | MR

[9] A. A. Samarskij, P. N. Vabishchevich, Vychislitel'naya teploperedacha, Knizhnyy dom “LIBROKOM” Publ., M., 2009, 784 pp. (In Russ.) | MR

[10] A. A. Samarskij, E. S. Nikolaev, Metody resheniya setochnyh uravnenij, Nauka Publ, Moscow, 1978, 592 pp. (In Russ.)